How to render a large number of stacked glass sheets
FIRSTLY, do you have photo reference? ;-) This render might actually BE really realistic, but your eye is fooled because your brain knows that it's fake. Also, photo reference will help you be specific about what changes to make.

The basic glass and transparent shaders aren't terribly realistic by default. You'll want to look into Physically Based glass shaders. It's a complicated subject, especially for Blender, but essentially you'll build your own shader with a transparent shader, a refraction shader (it doesn't do any reflection), and a glossy shader. You'll put these individual components together with mix shaders, modulating the factor on them with Fresnel or Layer Weight input.

The full details are a bit beyond the scope of this site, but searching for "physically based glass" should get you closer. Also looking at how people have built "fresnel glossy" shaders for Blender will be helpful because the principle is similar.

Here's a helpful link, for as long as it's available: http://www.chocofur.com/6-shadersamptextures.html

It would also be helpful to add a colored checker pattern and/or noise pattern behind and under the glass just for production, so you can see what's going on a little more easily. Then remove it once you get everything right. These stark white, grayscale environments can be very tricky.

Hope that helps!